of them there were either 
who were all friends among each other or 
and 
play a game, given an integer 
,
first, then every player sees the last number written and if it is 
then in his turn he writes 
or 
,
be the circumcircle of a scalene triangle 
be a circle internally tangent to 
and 
,
.
touch 
and 
,
and 
concur on the bisector of 
.
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
exist that satisfy 
for all real 
and the domain of f is strictly within the set of 
,
?
with edge 
cm.
and tangent to the plane 
.
)
and a plane 
,
and 
.
.
and 
,
and 
.
.
be the midpoint 
.
such that 
and 
. Prove that line 
passes through the midpoint of segment 
.
,
and 
are all irrational numbers.
over the interval ![$[-1, 1]$](http://latex.artofproblemsolving.com/5/c/3/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png)
?
Y by
Let 
be a triangle with 
, 
, 
. Let 
, and 
be points lying on sides 
, respectively. Given that 
, and 
are cyclic respectively, and the circumcircle of 
are tangent to line 
, find the length of segment 
.
be a triangle with 
, 
, 
. Let 
, and 
be points lying on sides 
, respectively. Given that 
, and 
are cyclic respectively, and the circumcircle of 
are tangent to line 
, find the length of segment 
.This post has been edited 3 times. Last edited by NicoN9, Apr 20, 2025, 11:22 AM
Reason: typo fixed
Reason: typo fixed
